Apparently this was just released, but I can't see any sign up button or anything, so it must just be that they've just announced it.
Giz did mention a while back that the BBC was planning on something like this, and it seems to have come true, they've teamed up with Spotify, YouTube, and Deezer (Who is Deezer?) to make this happen.
Well I can't wait to get to using it, I was thinking about signing up to Spotify recently but I haven't looked into it properly, and I think the way that this works is that you listen to music on iPlayer, click the "add to playlist" button, and it'll show up on Spotify, YouTube, and 'Deezer'. Actually, this looks kinda confusing, but it doesn't seem available yet.
Edit: Looking into it, when you hear music on iPlayer/iPlayer Radio, you can add the tracks to a playlist that'll then sync up to Spotify/YouTube/Deezer, and there you can listen to the music in full.